The derivative of the given function y = (cos 2x)^x can be obtained by transforming the given expression into its logarithmic form.
y = (cos 2x)^x
ln y = x ln (cos 2x)
dy/y = x * -dx/2 sin 2x + ln (cos 2x) dx
dy/dx = -xy/2 sin 2x + ln (cos 2x). This is the final answer
